Generalized FaddeevVolkov models Rinat Kashaev University of Geneva - PowerPoint PPT Presentation

Generalized Faddeev–Volkov models Rinat Kashaev University of Geneva RAQIS’16 Geneva, August 22-26, 2016 Rinat Kashaev Generalized Faddeev–Volkov models

Motivation: special functions and integrable lattice models with infinitely many local states Notable examples The quantum dilogarithm: the Faddeev–Volkov model (Bazhanov–Mangazeev–Sergeev) The elliptic beta function: a master solution of the quantum Yang–Baxter equation (Spiridonov, Bazhanov–Sergeev) Rinat Kashaev Generalized Faddeev–Volkov models

Review of Faddeev’s quantum dilogarithm ∞ 1 + e 2 π bx + π i b 2 (2 k +1) � Im( b 2 ) > 0 Φ b ( x ) := 1 + e 2 π b − 1 x − π i b − 2 (2 k +1) , k =0 Integral representations e − i 2 xz �� � Φ b ( x ) = exp 4 sinh( zb ) sinh( zb − 1 ) z dz (Faddeev) R + i 0 � i � � d t � e b 2 t +2 π bx + 1 � Φ b ( x ) = exp log (Woronowicz) e t + 1 2 π R ⇒ extension to b ∈ C \ i R Rinat Kashaev Generalized Faddeev–Volkov models

Φ b ( z − i s ) s = b ± 1 Φ b ( z +i s ) = 1 + e 4 π sz , Functional equations: 2 m + 1 n + 1 ib − 1 , � � � � Poles: z = ib + m , n ∈ Z ≥ 0 2 2 Zeros = − Poles Unitarity: (1 − | b | ) Im b = 0 ⇒ | Φ b ( x ) | = 1 ∀ x ∈ R Φ b ( x )Φ b ( − x ) = Φ b (0) 2 e i π x 2 Inversion relation: Pentagon identity: Φ b ( p )Φ b ( q ) = Φ b ( q )Φ b ( p + q )Φ b ( p ) 1 for self-adjoint Heisenberg operators pq − qp = (2 π � = 1) 2 π i Rinat Kashaev Generalized Faddeev–Volkov models

The weight function of the Faddeev–Volkov model W FV ( x , y ) = Φ b ( x − y ) Φ b ( x + y ) e 2 π i xy Symmetry properties: 1 W FV ( x , y ) = W FV ( x , − y ) = W FV ( − x , y ) The Yang–Baxter relation: W FV ( p , x ) W FV ( q , x + y ) W FV ( p , y ) = W FV ( q , y ) W FV ( p , x + y ) W FV ( q , x ) Rinat Kashaev Generalized Faddeev–Volkov models

Generalization to arbitrary self-dual LCA groups Let A be a LCA (locally compact abelian) group with the (Pontryagin) dual group ˆ A = all continuous group homomorphisms from A to the complex circle group T := { z ∈ C : | z | = 1 } . A is self-dual if there exists a group isomorphism f : A → ˆ A . Assume that there exists a function �·� : A → T (to be called gaussian exponential) such that � x ; y � := f ( x )( y ) = � x + y � � x � = �− x � , � x �� y � � Normalized Haar measure d µ ( x ): A 2 � x ; y � d µ ( x ) d µ ( y ) = 1 Examples 1 A = R , � x � = e π i α x 2 , � x ; y � = e 2 π i α xy , d µ ( x ) = � | α | d x 2 A = Z / N Z , � m � = e π i M N m ( m + N ) , � m ; n � = e 2 π i M N mn , √ d µ ( m ) = 1 / N 3 A = T × Z , � ( z , m ) � = z ± m , � ( u , m ); ( v , n ) � = ( u n v m ) ± 1 , d µ ( e 2 π i t , m ) = d t Rinat Kashaev Generalized Faddeev–Volkov models

The quantum dilogarithm over a self-dual LCA group Let A be a self-dual LCA group with a gaussian exponential � . � . The Fourier transformation operator F in L 2 ( A ) is defined by the integral kernel � x | F | y � = � x ; y � . To any function g : A → C , associate operators g ( q ), g ( p ) and g ( p + q ) in L 2 ( A ): ∀ f ∈ L 2 ( A ) , ( g ( q ) f )( x ) = g ( x ) f ( x ) , g ( p ) := F g ( q ) F − 1 , g ( p + q ) := � q � − 1 g ( p ) � q � . Definition (Andersen–K) A quantum dilogarithm over ( A , � . � ) is a function φ : A → T such that φ ( x ) φ ( − x ) = φ (0) 2 � x � , (inversion relation) ∀ x ∈ A , (pentagon relation) φ ( p ) φ ( q ) = φ ( q ) φ ( p + q ) φ ( p ) . Rinat Kashaev Generalized Faddeev–Volkov models

Remark The pentagon relation is equivalent to the integral relation � φ ( x ) φ ( y ) = A 3 D ( u , x , v , y , w ) φ ( w ) φ ( v ) φ ( u ) d µ ( u , v , w ) D ( u , x , v , y , w ) := γ � u − x ; w − y � � u − v + w � � γ := � z � d µ ( z ) A Rinat Kashaev Generalized Faddeev–Volkov models

Examples 1 A = R , � x � = e π i x 2 , φ ( x ) = Φ b ( x ), (1 − | b | ) Im b = 0 2 A = T × Z , � ( z , m ) � = z m φ ( z , m ) = z max( m , 0) 1 1+ q 1 − m +2 j z φ ( z , m ) = � ∞ 1+ q 1 − m +2 j / z , q ∈ ] − 1 , 1[, related to the 2 j =0 tetrahedron index of Dimofte–Gaiotto–Gukov 3 A = R × Z / k Z , � ( x , m ) � = e π i x 2 e − π i m ( m + k ) / k , k − 1 � φ ( x , m ) = Φ e i θ ( a j ( x , m )) j =0 + (1 − k ) cos θ + je − i θ � j + m � x − i e i θ a j ( x , m ) := √ i k k k θ ∈ [0 , π/ 2[, { x } := the fractional part of x . Rinat Kashaev Generalized Faddeev–Volkov models

Generalized Faddeev–Volkov weight function Let φ ( x ) be a quantum dilogarithm over a self-dual LCA group ( A , � . � ). The associated Faddeev–Volkov weight function is defined by the formula w( x , y ) := φ ( x − y ) φ ( x + y ) � x ; y � Symmetry properties 1 w( x , y ) = w( x , − y ) = w( − x , y ) Yang–Baxter relation w( p , x ) w( q , x + y ) w( p , y ) = w( q , y ) w( p , x + y ) w( q , x ) Rinat Kashaev Generalized Faddeev–Volkov models

The standard interpretation as a star-triangle relation � w( u − t , x ) w( t , x + y ) ˜ ˜ w( t − s , y ) d µ ( t ) A = w( u , y ) ˜ w( u − s , x + y ) w( s , x ) � w( y , x ) := ˜ � y ; z � w( z , x ) d µ ( z ) A Rinat Kashaev Generalized Faddeev–Volkov models

Weil transformation Let B ⊂ A be a closed subgroup such that B = B ⊥ := { x ∈ A | � x ; b � = 1 , ∀ b ∈ B } We have a group homomorphism p : A → ˆ B , p ( x )( b ) = � x ; b � , inducing a natural group isomorphism A / B ≃ ˆ B . Define the Weil transformation of w( x , y ): � w( s , t , x ) := ˇ w( s + b , x ) � t ; b � d µ ( b ) B The inverse Weil transformation � w( s , x ) = w( s , t , x ) d µ ( t ) ˇ A / B Quasi B -periodicity properties w( s + b , t , x ) = �− b ; t � ˇ ˇ w( s , t , x ) , w( s , t + b , x ) = ˇ ˇ w( s , t , x ) w( s , t , x ) is a section of a complex line bundle over ( A / B ) 2 . ⇒ ˇ Rinat Kashaev Generalized Faddeev–Volkov models

Interpretation as IRF-model with abelian gauge symmetry Let χ : A 2 → T be a bi-character such that � x ; y � = χ ( x , y ) χ ( y , x ) . Then M( x , y , z ) := χ ( x , y ) ˇ w( x , y , z ) is quasi B -periodic M( x , y + b , z ) = χ ( x , b ) M( x , y , z ) , M( x + b , y , z ) = ¯ χ ( y , b ) M( x , y , z ) and satisfies a Yang–Baxter relation of IRF-type c c b b α α � � g a d µ ( g ) = a d µ ( h ) d d h A / B A / B β β e e f f γ γ b := M( a − c , d − b , α − β ) a c d α β Rinat Kashaev Generalized Faddeev–Volkov models

Example: the IRF-form of the Faddeev–Volkov model � x � = e π i x 2 , A = R , φ ( x ) = Φ b ( x ) A / B = R / Z ≃ T ≃ ˆ Z = ˆ B = Z ⊂ R = A , B � x ; y � = e 2 π i xy , χ ( x , y ) = e π i xy M( x , y , z ) φ (0) 2 � 1 = e π i( xy − x 2 − z 2 ) φ ( x − z , u + y − x − 1 / 2)ˇ ˇ φ ( − x − z , u ) d u 0 with the Weil–Gelfand–Zak transformation of the quantum dilogarithm ˇ � φ ( x + k ) e 2 π i ky φ ( x , y ) = k ∈ Z Rinat Kashaev Generalized Faddeev–Volkov models

The special value b = e π i / 6 � − e 2 π (¯ b x − i y ) � − e 2 π i x ; q � � ( q ; q ) ∞ ∞ θ q ˇ φ ( x , y ) = � � − e 2 π ¯ � � ( e − 2 π i y ; q ) ∞ − e 2 π i( x + y ) ; q b x ∞ θ q √ where q := e 2 π i b 2 = i e − π 3 , ∞ � (1 − xq n ) , ( x ; q ) ∞ := n =0 k 2 2 x k = ( q ; q ) ∞ ( −√ qx ; q ) ∞ ( −√ q / x ; q ) ∞ � θ q ( x ) := q k ∈ Z Rinat Kashaev Generalized Faddeev–Volkov models